Complexities of some interesting problems on spanning trees

1. Complexities of some interesting problems on spanning trees M Sohel Rahman King’s College, London M Kaykobad KHU, NSU and BUET

2. Abstract • Complexity issues of some interesting spanning tree problems by imposing various constraints and restrictions on graph parameters. • Introduce a new notion of “set version” of a problem by replacing bounds by a set of that cardinality. • Maximum leaf spanning tree is one such example

3. Problems under consideration • Problem 1.1(Degree constrained spanning tree): Given a connected graph G=(V,E) and a positive integer K<|V|, we are asked the question whether there is a spanning tree of G such that no vertex in T has degree larger than K. • Theorem 1.2: degree constrained spanning tree problem is NP-Complete.

4. Problems (contd.) • Problem 1.3 (maximum Leaf Spanning Tree Problem): Given a connected graph G=(V,E) and a positive integer K<|V|, we are asked the question whether there is a spanning tree of G such that K or more vertices in T have degree 1. • Theorem 1.4 Maximum Leaf Spanning Tree Problem is NP-Complete.

5. New Problems • We denote by NG(x)- the set of vertices adjacent to vertex x, dG(x) its cardinality. • Subgraph of G induced by a set S of vertices is denoted by <S> • ΠG={v|v is a leaf in G} • Matching M of G from A to B⊆V none of A or B has degree more than 1

6. New problems and results • Problem 2.1 (Minimum Leaf Spanning Tree): Given a connected graph G=(V,E) and a positive integer K<|V| we are asked the question whether there is a spanning tree T of G such that K or less vertices have degree 1. • Theorem 2.2 Minimum Leaf Spanning Tree Problem is NP-Complete.

7. New Problems and Results(contd.) • Problem 2.3 (Restricted-Leaf-in-Subgraph Spanning Tree Problem): Given G=(V,E) be a connected graph, X a vertex subset of G and a positive integer K<|X|, we are asked the question whether there is a spanning tree TG such that number of leaves in TG belonging to X is less than or equal to K.

8. New problems and Results(contd.) • Theorem 2.4 Restricted-Leaf-in-Subgraph Spanning Tree Problem is NP-Complete. • Proof: If X=V then it is Minimum Leaf Spanning Tree Problem. Hence it is NP-Complete. • Now we consider a variant of Maximum Leaf Spanning Tree for Bipartite Graphs.

9. New Problems and Results(contd.) • Problem 2.5(variant of Maximum Leaf Spanning Tree for Bipartite Graphs) Let G be a connected bipartite graph with partite sets X and Y. Given a positive integer K<=|X| we are asked the question whether there is a spanning tree TG in G such that the number of leaves in TG belonging to X is greater than or equal to K.

10. New Problems and Results(contd.) • Theorem 2.6. Let G be a connected bipartite graph with partite sets X and Y and suppose K is a positive integer such that K<=|X|. Then there is a spanning tree T in G such that the number of leaves in T belonging to X is grater than or equal to K iff there is a set S⊆X such that |X-S|>=K and <S∪Y> is connected.

11. New Problems and Results(contd.) • Theorem 2.7 Let G be a connected bipartite graph with partite sets X and Y and suppose K is a positive integer such that K<=|X|. Then there is a spanning tree T in G such that the number of leaves in T belonging to X is greater than or equal to K iff there is a set S⊆X such that all the followings hold true: a) |X-S|>=K b) <S∪Y> is connected c) for any subset S’⊆S |NG(S’)|>=|S’|+1

12. New Problems and Results(contd.) • Problems of set version • Problem 3.1 (Set Version of Maximum Leaf spanning Tree problem) Given a connected graph G=(V,E) and X⊆V, we are asked the question whether there is a spanning tree T such that X⊆ПT, where ПT={v|v is a leaf of T}

13. New Problems and Results(contd.) • Theorem 3.2 Let G=(V,E) be a connected graph, X⊆V and Y=V-X. Then there exists a spanning tree T such that X⊆ПT, if and only if both of the following conditions hold true: • 1) <Y> is connected, • 2) Every X-node has an adjacent node in Y.

14. New Problems and Results(contd.) • Theorem 3.3 Set version of the Maximum Leaf Spanning Tree problem is polynomially solvable. • Problem 3.4 (Set version of Problem 2.5) Let G be a connected bipartite graph with partite sets X and Y and X1 ⊆X. we are asked the question whether there is a spanning tree TG in G such that X1 ⊆ ПT , where ПT={v|v is a leaf of T}

15. New Problems and Results(contd.) • Theorem 3.4 Problem 3.4 is polynomially solvable. • Problem 3.6( Set version of Minimum Leaf Spanning Tree problem): Given a connected graph G=(V,E) and X⊆V, we are asked the question whether there is a spanning tree T such that ПT⊆ X, where ПT={v|v is a leaf of T}

16. New Problems and Results(contd.) • Theorem 3.7 Set version of Minimum Leaf Spanning Tree Problem is NP-Complete. • References • EW Dijkstra, Self-stabilizing systems in spite of distributed control, ACM 17(1974) 643-644 • MR Garey, DS Johnson, Computers and Intractability, Freeman, New York, 1979 • P Hall, Representation of subsets, J London Math Soc 10(1935) • M Sohel Rahman, M Kaykobad, Complexities of some interesting problems on spanning trees, Information processing Letters 94(2005)93-97

17. THANK YOU VERY MUCH